Approximating the number of integers without large prime factors

Abstract

Ψ ( x , y ) \Psi (x,y) denotes the number of positive integers ≤ x \leq x and free of prime factors > y >y . Hildebrand and Tenenbaum gave a smooth approximation formula for Ψ ( x , y ) \Psi (x,y) in the range ( log ⁡ x ) 1 + ϵ > y ≤ x (\log x)^{1+\epsilon }> y \leq x , where ϵ \epsilon is a fixed positive number ≤ 1 / 2 \leq 1/2 . In this paper, by modifying their approximation formula, we provide a fast algorithm to approximate Ψ ( x , y ) \Psi (x,y) . The computational complexity of this algorithm is O ( ( log ⁡ x ) ( log ⁡ y ) ) O(\sqrt {(\log x)(\log y)}) . We give numerical results which show that this algorithm provides accurate estimates for Ψ ( x , y ) \Psi (x,y) and is faster than conventional methods such as algorithms exploiting Dickman’s function.